Bayesian networks in additive manufacturing and reliability engineering

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Azarakhsh Hamedi

BAYESIAN NETWORKS IN ADDITIVE MANUFACTURING AND RELIABILITY ENGINEERING

Automation Technology and

Mechanical Engineering

Master of science thesis

March 2019

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ABSTRACT

Azarakhsh Hamedi: Bayesian networks in additive manufacturing and reliability engineering Master of Science Thesis

Tampere University

Automation Engineering, Factory Automation and Industrial Informatics February 2019

A Bayesian network (BN) is a powerful tool to represent the quantitative and qualitative features of a system in an intuitive yet sophisticated manner. The qualitative aspect is represented with a directed acyclic graph (DAG), depicting dependency relations between the random variables of the system. In a DAG, the variables of the system are shown with a set of nodes and the dependencies between them are shown with a directed edge. A DAG in the Bayesian network can be a causal graph under certain circumstances. The quantitative aspect is the local conditional probabilities associated with each variable, which is a factorization of the joint probability distribution of the variables in the system based on the dependency relation represented in the DAG.

In this study, the benefits of using BNs in reliability engineering and additive manufacturing is investigated. In the case of reliability engineering, there are several methods to create predictive models for reliability features of a system. Predicting the possibility and the time of a possible failure is one of the important tasks in the reliability engineering principle. The quality of the corrective maintenance after each failure is affecting consecutive failure times. If a maintenance task after each failure involves replacing all the components of an equipment, called perfect maintenance, it is considered that the equipment is restored to an “as good as new” (AGAN) condition, and based on that, the consecutive failure times are considered independent. Not only in most of the cases the maintenance is not perfect, but the environment of the equipment and the usage patterns have a significant effect on the consecutive failure times. In this study, this effect is investigated by using Bayesian network structural learning algorithms to learn a BN based on the failure data of an industrial water pump.

In additive manufacturing (AM) field, manufacturing systems are normally a complex combination of multiple components. This complex nature and the associated uncertainties in design and manufacturing parameters in additive manufacturing promotes the need for models that can handle uncertainties and are efficient in calculations. Moreover, the lack of AM knowledge in practitioners is one of the main obstacles for democratizing it. In this study, a method is developed for creating Bayesian network models for AM systems that includes experts’ and domain knowledge.

To form the structure of the model, causal graphs obtained through dimensional analysis conceptual modeling (DACM) framework is used as the DAG for a Bayesian network after some modifications. DACM is a framework for extracting the causal graph and the governing equations between the variables of a complex system. The experts’ knowledge is extracted through a probability assessment process, called the analytical hierarchy process (AHP) and encoded into local probability tables associated with the independent variables of the model. To complete the model, a sampling technique is used along with the governing equations between the intermediate and output variables to obtain the rest of the probability tables.

Such models can be used in many use cases, namely domain knowledge representation, defect prognosis and diagnosis and design space exploration. The qualitative aspect of the model is obtained from the physical phenomena in the system and the quantitative aspect is obtained from the experts’ knowledge, therefore the model can interactively represent the domain and the experts’ knowledge. In prognosis tasks, the probability distribution for the values that an output variable can take is calculated based on the values chosen for the input variables. In diagnosis tasks, the designer can investigate the reason for having a specific value in an output variable among the inputs. Finally, the model can be used to perform design space exploration. The model reduces the design space into a discretized and interactive Bayesian network space which is very convenient for design space exploration.

Keywords: additive manufacturing, Bayesian networks, causal models, reliability engineering The originality of this thesis has been checked using the Turnitin Originality Check service.

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PREFACE

This work is a result of an 8-months research in the Additive Manufacturing research group in the Automation Technology and Mechanical Engineering (ATME) (former Mechanical engineering and industrial systems, MEI) laboratory in Tampere University (Former Tampere University of Technology).

I would like to reflect my gratitude to Eric Coatanéa and Jouko Laitinen for granting me the chance to be part of their team in the group. My utmost appreciation goes to Eric Coataéa for his supervision and support.

This work is also the result of a close teamwork in the additive manufacturing research group. I would like to specially thank Hossein Mokhtarian for his help and guidance during this study.

I would like to express my thanks to Jose M. L. Lastra and Anderi Lobov and the other staff in FAST-Lab for giving me the opportunity to learn and guiding me through my studies.

I would also like to thank Sara Talebian and Amir Dirin for their endless care during these last years. Finally, I owe my heartfelt thanks to my parents, Dadar Hamedi and Shahla M. Hosseini, for their continuous encouragement throughout my life.

Stockholm, 18 March 2019

Azarakhsh Hamedi

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LIST OF FIGURES

Figure 1. SLM process (Proform, 2018) ... 12

Figure 2. Material Extrusion Additive Manufacturing (Gonzalez-Gutierrez et al., 2018) ... 13

Figure 3. Powder bed fusion (Frazier, 2014) ... 14

Figure 4. A generic powder and electron beam DED system (Frazier, 2014) ... 15

Figure 5. Wire and Arc Additive manufacturing (McAndrew et al., 2018) ... 16

Figure 6. Curling defect in overhanging parts (Tounsi & Vignat, 2017) ... 17

Figure 7. An overhanging part with a support structure (Tounsi & Vignat, 2017)... 17

Figure 8. Perception of degradation, diagnostics and prognostics in health management (Lee et al., 2014) ... 18

Figure 9. Perfect, imperfect and minimal maintenance and their effect on the failure rate (Carlo & Arleo, 2017) ... 21

Figure 10. A simple Hidden Markov Model ... 37

Figure 11. Missing value mechanisms (Conrady & Jouffe, 2007) ... 53

Figure 12. A tree level hierarchy (Saaty & Vargas, 2012) ... 56

Figure 13. Fundamental variables and their interconnections in the bond graph context (Mokhtarian, Coatanéa, & Paris, 2017) ... 61

Figure 14. DACM modelling approach (Mokhtarian, Coatanéa, Paris, Mbow, Pourroy, Marin, & Ellman, 2018) ... 62

Figure 15. Causal ordering algorithm (Mokhtarian et al., 2017) ... 65

Figure 16. The workflow for creating a Bayesian network using DACM Framework’s outputs ... 67

Figure 17. Sample Causal Graph Created with DACM Framework ... 68

Figure 18. The network after removing Exogenous variables ... 69

Figure 19. Sample graph after adding intermediate variables... 70

Figure 20. The graph after adding ratio constraints ... 73

Figure 21. Dependent 5 and its two parent nodes, independent 5 and independent 1. ... 74

Figure 22. The process of finding the range of the dependent variable ... 74

Figure 23. Sampling method for finding probability distribution of dependent nodes ... 75

Figure 24. The functional model for the cantilever part manufactured with curling defect, updated from Mokhtarian et al. (2018) ... 90

Figure 25. The causal graph obtained from the functional model and the governing equation Mokhtarian et al. (2018) ... 93

Figure 26. The causal graph produced by DACM Framework for curling defect in PBF ... 94

Figure 27. The work flow for creating a Bayesian network using DACM as described in section 4.1.2... 94

Figure 28. The graph after removing exogenous variables and adding intermediate variable “Moment of inertia of supports” ... 95

Figure 29. The final Bayesian network structure for the curling defect case study ... 99

Figure 30. The process and timeline of data collection ... 103

Figure 31. The pump failure timelines after alignment ... 103

Figure 32. The failure timeline for an instance of the pump ... 104

Figure 33. The SC and CTF curve ... 109

Figure 34. BN structure learned from the dataset ... 110

Figure 35. Monitor screens for TTF variables ... 111

Figure 36. Monitor screens for CF variables ... 111

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independent and performance variables ... 114 Figure 38. Changes in Curling defect and Mass of the supports due to

changes in the number of supports ... 116 Figure 39. Diagnosing the reason for having a curling defect between 2mm to

10mm. ... 118 Figure 40. The monitors for the model with the initial parameter design

configuration ... 119 Figure 41. The monitors for design variables after choosing more design

variables ... 120 Figure 42. Posterior distributions after setting a target value of having less

than 0.2mm of defect ... 122 Figure 43. The BN showing the dependencies between TTF and CF variables... 123 Figure 44. Monitor windows after setting evidence for the failures ... 125

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LIST OF SYMBOLS AND ABBREVIATIONS

ABS Acrylonitrile Butadiene Styrene

ABAO As Bad as Old

AGAN As Good as New

AHP Analytical Hierarchy Process AI Artificial Intelligence

AIC Akaike Information Criterion

AM Additive Manufacturing

ASM Aggressive Space Mapping

ASTM American Society for Testing and Materials BDc Bayesian Dirichlet Criterion

BIC Bayesian Information Criterion

CAD Computer Aided Design

CBM Condition Based Maintenance

CEN European Committee for Standardization

CF Censored Failure

CFPR Carbon Fibre Reinforced Plastic

CFR Constant Failure Rate

CM Corrective Maintenance

CI Consistency Index

CPT Conditional Probability Table

CR Consistency Ratio

CTF Contingency Table Fit

DA Dimensional Analysis

DACM Dimensional Analysis Conceptual Modelling

DAG Directed Acyclic Graph

DBN Dynamic Bayesian Network

DC Direct Current

DED Direct Energy Deposition

DFR Descending Failure Rate

DFX Design for X

DSE Design Space Exploration

EBM Electron Beam Melting

EM Expectation Maximization

FDM Fused Deposition Modelling

FE Finite Element

FFF Fused Filament Fabrication

FMECA Failure Modes and Effective Critically Analysis FPBN Fault Predicting Bayesian Network

FTA Fault Tree Analysis

HMM Hidden Markov Model

ICME Integrated Computational Materials Engineering IFR Increasing Failure Rate

IPD Interaction Preserving Discretizations

ISO International Organization for Standardization JPD Joint Probability Distribution

KL Kullback-Leibler

MAR Missing at Random

MCAR Missing Completely at Random

MDL Minimum Description Length

MEAM Material Extrusion Additive Manufacturing

MI Mutual Information

MML Minimum Message Length

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MPT Marginal Probability Tables

NIST National Institute of Standardization

NMAR Not Missing at Random

PHM Prognosis and Health Management

PLA Poly Lactic Acid

PM Preventative Maintenance

RBD Reliability Block Diagram

RCM Reliability Centred Maintenance

RI Random Consistency Index

RUL Remaining Useful Lifetime

SADT Structured Analysis and Design Technique

SC Structural Coefficient

SLA Stereo Lithography

SLM Selective Laser Melting SLS Selective Laser Sintering

SSM State Space Models

TTF Time to Failure

TTTF Total Time to Failure

VP Vat Photo-Polymerization

WAAM Wire and Arc Additive Manufacturing

𝛂 Thermal expansion

σ Thermal constraint

δ Curling defect

F Force

m Mass

a Acceleration

θ Temperature

p Pressure

ρ Material Density

q Heat Energy Input

k Coefficient of conduction

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1. INTRODUCTION

Limited resources and fierce competition in the market encourages the manufacturers to adopt new manufacturing technologies which are more flexible, more predictable, more agile and needs less preparation time (d’Aveni, 2015). This urges manufacturers to use more complex manufacturing equipment and processes. Handling this complexity re- quires more knowledge and sophisticated methods and models.

One of the consequences of using more complex systems and models is facing uncertainty in the system (de Rocquigny, Devictor, & Tarantola, 2008). Uncertainty in manufacturing systems may have different source and it is classified by Nannapaneni et al.

(2016) three categories. Uncertainty may be because of the quality of the data, e.g. in- adequate, missing, erroneous data. Another category of uncertainty occurs because of assumptions and approximations in the models used. These two types of uncertainty happened because of lack of knowledge and called epistemic uncertainty. The third category of uncertainty is because of natural varieties in the manufacturing process and called statistical or aleatory uncertainty.

Uncertainty shows itself in industrial practice in different situations. As Rocquigny et al.

(2008) discuss, uncertainty may occur because of variability or error in measurements in variables, having an expected value for a variable, having confidence intervals for some variables, variables relating to the risk percentage, having probability of exciding a threshold or having ranges for variables in the design phase. Some areas such reliability of the equipment are in direct relation with uncertainties in the system (O’Connor &

Kleyner, 2012).

On the other hand, one of the obstacles to using new complex equipment and systems in manufacturing is the lack of expert’s knowledge of using those processes among designers and practitioners. Creative design and manufacturing with new technologies like additive manufacturing need special tools, knowledge, and expertise which is sparse due to the recentness of these technologies (Gardan, 2014). New concepts such as design for X (DFX) combines the state of the art models and the expert’s knowledge of manufacturing equipment and processes to provide interactive tools for designers. Such systems enable designers to maximize their creativity in the early design stage while the

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system can show the result of their choices in real-time with some degree of uncertainty (Laverne, Segonds, D’Antonio, & Le Coq, 2017).

One of the of the approaches to creating a model that can handle the uncertainty includes experts’ knowledge, provides an interactive interface and is efficient in computation is using Bayesian networks. There are several other approaches such as fuzzy logic, neu- ral networks, and rule-based expert systems but none of them can handle all of the mentioned criteria at the same time (McNaught & Chan, 2011).

The need for bidirectional inferable models, i.e. Semantic (up-down evidence reading) and perceptual (down-up evidence reading) inferable models, leads to initial deployment of the Bayesian networks. A Bayesian network is a graphical probabilistic model that represents a qualitative and a quantitative relationship between a set of random variables. The qualitative part is described using directed acyclic graphs (DAG) to show the dependencies between random variables and the quantitative part is the probabilistic relationships between those variables. The quantitative part is based on local probability distributions between the random variables and it represents a particular factorization of the joint probability distribution of the variables based on the relations specified through the DAG (Pearl, 2004).

In this representation, each random variable is represented with a node or vertex in the graph and a directed arc, also called an arrow or an arc, from node 𝐴 to 𝐵 shows that node 𝐵 is dependent to 𝐴 and 𝐴 is possibly a cause for the node 𝐵. Since there can be many different factorizations for a joint distribution, there can be as many BNs for the same distribution. A fully connected network is the best realization of the joint probability distribution in the form of a Bayesian network. The missing arcs between nodes is a valuable information in a BN. They represent the conditional independence between the random variables and help representing the joint distribution in a more compact form (Judae Pearl, 1988).

The other benefit of using a Bayesian network is that using it, it is possible to encode the expert’s knowledge into a model. The experts’ knowledge can be extracted in the form of the dependency relations between variables, i.e. the structure of the network, or the quality of interactions between variables, i.e. the probability tables (Williamson, 2001).

After creating the model, using the DAG and the probability tables, it is possible to perform Bayesian inference between the variables of the model. The inference process cal- culates the effect of changes in the probability distribution of one or several nodes on the probability distributions of the other nodes. Several inference algorithms have been developed which can perform this task efficiently. This enables the Bayesian networks to

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be not only fast and efficient in calculations, but also have an interactive nature (Guo &

Hsu, 2002).

1.1 Research Objective and Scope

This study tries to investigate the possibility of using a systematic approach to create models for the uncertainty associated problems in the industrial domain using Bayesian networks.

There are two general approaches for creating Bayesian network models for a problem in a system, namely the knowledge elicitation approach and the machine learning approach. BNs can be obtained in a subjective manner by eliciting experts knowledge and the domain knowledge for the dependency of the variables and the probability distributions (Koller, Friedman, Getoor, & Taskar, 2007). Multiple methods have been developed to obtaining expert’s and domain knowledge for the structure of the Bayesian networks (K.W. Przytula & Thompson, 2002; Richardson & Domingos, 2003) and the correspond- ing probability distributions (Nunes et al., 2018).

The other method of creating a Bayesian network model is to use the available data in the domain and obtain a Bayesian network using machine learning algorithms. Although several methods have been developed to perform the machine learning (Daly, Shen, &

Aitken, 2009) which are quite effective and efficient, the main problem is acquiring suitable data and preparations of the data to be used in the machine learning process (sections 3.1.5-3.1.7).

Therefore, the first objective of this research is to develop a systematic method to create interactive Bayesian network models for complex systems in order to predict the results of the choice of design and manufacturing parameters in the early stage design phase.

The second objective of this research is to create a predictive Bayesian network model for an industrial problem using the data and machine learning to get familiar with the challenges and develop a systematic approach for similar problems.

The methods that are created and gathered in this study are implemented in two industrial case studies. A problem in an additive manufacturing system is chosen to be modelled with a BN model using experts’ and domain’s knowledge and an equipment reliability case study is chosen to be modelled with BNs using machine learning and data.

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1.2 Case Studies

One of the emerging technologies for which helps to address the needs in today’s fast pacing manufacturing is additive manufacturing (AM). AM is the process of manufacturing of parts by adding materials is layer by layer directly from their digital blueprints (d’Aveni, 2015). There are several obstacles in integrating AM technologies into production systems, one of which is the high degree of complexity in AM systems. Such systems are created by adjoining several complex subsystems and this makes it very difficult to create a holistic model for them to predict and assure the quality of the manufactured parts (Kathryn et al., 2016).

The other reason for the complexity of AM processes is the sheer number of input variables and that a big portion of the processes within the system are not identified. There- fore, finding the right parameters for the system to reach desirable dimensional, mechanical and metallurgical characteristics is a multi-criteria problem (Stavropoulos &

Foteinopoulos, 2018). Moreover, the other major challenges in democratizing AM is lack of knowledge and expertise of AM among designers and practitioners (Lindqvist, Piili, &

Salminen, 2016), therefore the models should be able to contain and represent experts’

knowledge in the field.

One of the challenges in manufacturing using AM is the defects in the manufactured parts. Additive manufacturing process causes a number of defects in the parts and since the process is complicated and fast pacing, it is hard to create exact models for them (Mindt, Desmaison, Megahed, Peralta, & Neumann, 2017). Moreover, choices in design and manufacturing parameters have a significant effect on the extent of these defects.

Therefore, there is a significant amount of uncertainty associated with the variables of the system (Béraud, Vignat, Villeneuve, & Dendievel, 2014).

The other case study in this thesis is addressing the failure prediction in reliability engineering principle. Failure in manufacturing equipment imposes costs to the production.

These costs can be the cost of downtime, excess maintenance, lost production, equipment repair, equipment replacement, and safety risks. These can affect companies in short or mid-terms and it can even lead to loss of business in the long term. Taking advanced maintenance policies can reduce cost and risk significantly. Manufacturers can take advantage of preventive or planned maintenance by creating predictive models of the failures from the historical data of their equipment components (Letot, Equeter, Dutoit, & Dehombreux, 2017). Using such models and considering the current situation of the machinery, the optimal time of maintenance of the system can be predicted and costly failure can be prevented.

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Modelling the mechanism of failure is necessary to perform preventative and predictive maintenance. The well-known parametric models can describe the systems with good accuracy, but they lack the ability to adopt the changes in multiple variables of the model at the same time (Langseth, 1998).

The quality of each maintenance procedure, which is taking place after each failure, is affecting the expected time to the next failure. The maintenance quality can be ranging from perfect, i.e. bringing the equipment back to the “as good as new” condition, to a maintenance that makes the equipment’s health even worse than before the maintenance. The other factor which is important in the reliability of the equipment is the usage of the equipment and the environment of the operation (Carlo & Arleo, 2017). All these factors should be considered when a model is created for the reliability of equipment.

One of the major issues is the quality of the data in real-world cases. In the field of prediction and health management in reliability engineering, the data for the health condition of the equipment is very hard to find, partly due to the privacy policy of the companies and partly due to the nature of such systems. Field systems are typically not properly instrumented and the process of collecting data is time-consuming and expensive (Saxena, Goebel, Simon, & Eklund, 2008). The data used in this study is a single variable dataset of the failure times of industrial water pumps.

The other problem with the health condition data in the industrial domain is being subject to missingness and censoring. Missingness occurs when a data point is being failed to record 3.1.7). Censoring is a condition specific to failure data and it is basically the data which becomes invisible due to reasons such as ending the study or occurring before the study begins etc. (section 2.3.1).

1.3 Problem Definition

As mentioned before, this study tries to investigate the modelling process using BNs with two approaches in the industrial domain. The objectives of this research are implemented on the problems of the case studies. Therefore, the research questions of this study are:

How to model the probability of occurrence of a defect in an additive manufacturing process or:

• How to use the benefits of Bayesian networks in creating interactive models for curling defect problem in the additive manufacturing process which contains experts’ knowledge in the field of AM?

And the expected result in this field is:

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• To create a methodology for creating Bayesian network models for problems in complex systems in a systematic manner.

And the research questions in reliability engineering are:

• How to use machine learning in Bayesian networks for predicting failure times historical time to failure data?

Expected result for the case study in reliability engineering is:

• To extract a predictive model from data to estimate the next time to failure.

1.4 Methodology

To answer the first research question, this study tries to use dimensional analysis conceptual modelling (DACM) framework to obtain the structure of the Bayesian network and analytical hierarchy process (AHP) and a sampling technique to obtain the probability distributions.

DACM is proposing a series of methods to simplify, organize and simulate the behaviour of a system in the form of cause-effect relationships using qualitative information about that system. The result is a directed graph containing the causal relationships between the variables of the system and the governing equations between those variables (Coatanéa, Roca, Mokhtarian, Mokammel, & Ikkala, 2016).

AHP is initially developed as a method to derive priorities for different criteria in a multicriteria decision problem. AHP decomposes the criterion for decision problem into sub- criteria and acquires the expert’s preferences on those sub-criteria by performing two by two comparisons between them and finally synthesises a weight for each of them using special mathematical machinery (Saaty & Vargas, 2012).

To address the second research question, this study also tries to exploit the machine learning approach for obtaining a Bayesian network model for an industrial system using data. Several challenges are associated with the quality of the data in most of the industrial cases.

This study attempts to encounter the problems which are normally associated with datasets available in the field of reliability engineering in a systematic manner. Then a model for the problem is created using a machine learning technique and the model is validated against the dataset.

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1.5 Thesis Outline

This thesis is formed in five sections. After the introduction in this section, in the second section, the relevant background about Bayesian networks, additive manufacturing and reliability engineering is described briefly. This section creates the context of the case studies and shows the importance and the need for performing this study.

In the third section, the theoretical aspects of the methods used in the study are described in detail. The parts of Bayesian networks theory that are used in the study, the AHP and DACM methods which are used in developing the additive manufacturing case study and the methods developed for two case studies as well as the state of the art methods are described in this section.

The fourth section is dedicated to the details about the implementation of developed methodologies in the case studies. All steps are described in details and the resulting model is presented

In the fifth section, first, a brief discussion about the result of the case studies is presented and finally, the conclusion of the study is discussed.

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2. BACKGROUND

This section provides background about the Bayesian networks and the case studies of this thesis. First, the Bayesian network is introduced briefly and then some information regarding additive manufacturing systems and prognosis and health management is provided in order to set the context and show the importance of the research. The focus of the section is mostly on the case studies and the detail of the Bayesian network is described in section 3.1 in detail.

2.1 Bayesian networks

A Bayesian network (BN) is a graphical probabilistic model, which represents the qualitative and quantitative relationships of a set of random variables a single model. A directed acyclic graph (DAG), which is also called the structure of the BN, is illustrating the dependency the random variables. The random variables are shown with nodes and the dependencies between them are shown with a directed edge. In Bayesian networks the qualitative part, the DAG can be considered as a causal graph under certain circumstances (see section Backgrounds3.1.1).

The quantitative part of a BN is the conditional probability distributions of the set of random variables which their dependency relations are shown in the DAG. Having the DAG, the joint distribution of the random variables can be factorized into a multiplication of conditional probability distributions. This enables BNs to provide a compact representation of the joint probability distributions. In this representation, each random variable is represented with a node (vertices) and a directed arc (also called arrow of arc) from node A to B shows that node B is conditioned on node A in that particular factorization of that joint distribution. Since there can be many different factorizations for joint distribution, there can be as many BNs for the same distribution. The valuable information in a BN is the missing arcs between nodes. They are representing the conditional independence of random variables in that particular variable set (Ghahramani, 2001; McNaught & Chan, 2011).

The variables in a BN can be continuous, categorical, discrete valued or a combination of them. if the variables are continuing variables, the numerical values and their probability distribution functions are used and If they are categorical, intervals or discreet, they

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are described with categories or states and conditional probability tables (CPTs) (see section Backgrounds3.1.1).

Bayesian networks allow us to use the information from a subset of variables in the system to predict the behaviour of any other subset of variables in that system and make rational decisions according to that (Munteanu & Bendou, 2001).

The structure of a Bayesian network can be obtained using two general approaches. The first approach is trying to use machine learning techniques to learn the structure from the data recorded about the system previously. Therefore, the resulting network approxi- mates the joint probability of that dataset. Williamson (2001) calls these networks as abstract structures. The other approach is to have an interpretation of the Bayesian network in which the graph is representing a causal representation of the system and it may be subjective or objective. In the subjective case, the relation between two nodes, which is represented by a directed arc, is a direct causal relationship. In the objective case, this relation is the belief of an agent about the causal relationship between the variables of the system (nodes).

Advantages and uses of using Bayesian networks

Heckerman (1995) counted a few advantages of using Bayesian networks as follows.

First, handling incomplete data is a natural feature of Bayesian networks. Most of the other data analysis methods, e.g. regression and classification are prone to magnificent errors in case system variables are highly anti-correlated and for example, one of them is unobserved. Bayesian networks can encode statistical dependencies between variables, so they can handle incomplete data.

Secondly, using Bayesian networks, one can learn the causal relationship between variables in that domain. This can include valuable information about a system and the result can be utilized in other analysis methods. Moreover, using the causal network, it is possible to perform interventions and investigate the predicted results.

The third advantage is that Bayesian networks model domain knowledge and the data at the same time. Therefore, using the causal relationships in the Bayesian networks and Bayesian and non-Bayesian statistical tools makes a sophisticated package for data analysis.

Bayesian networks are used in several domains such as medical diagnosis, map learning, natural language processing, image processing, computational biology, civil infra- structure networks, epidemiology, etc. (Koller & Friedman, 2013).

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McNaught and Chan (2011) named a few uses of Bayesian networks in the industry as follows. Bayesian networks have been vastly used in fault diagnostics and failure prediction in manufacturing due to the uncertain nature of events in those principles. BNs are also used for reliability and risk assessment of manufacturing processes. The cause- effect modelling in BNs is a good tool to be used for manufacturing process scheduling under uncertainty. Also, BNs have been used in the field of predictive maintenance to determine the optimal time for a maintenance task to be performed. In a more general perspective, a set of Bayesian network models for different aspects of a factory have been combined to maximize the productivity of the factory. In a similar approach, BNs have been used as recommender systems to the customers of a customized manufacturing system to choose the best combinations.

2.2 Additive manufacturing

Additive Manufacturing (AM) is the process of joining material, layer by layer, Line by line or piece by piece, in order to fabricate a product directly from its digital 3D model. The term additive is used in opposition to subtractive manufacturing in which a product is created by subtracting material from a material block (ISO/ASTM, 2015).

As Yunlong and Yaoyao (2015) stated, additive manufacturing has three main advantages to previous methods. First, the production of highly complex parts can be done in a single process and the manufacturing cost will not increase with the complexity.

Secondly, multi-material parts with complex material combinations can be produced eas- ily with this method. And finally, manufacturing preparation time can be significantly de- creasing since parts can be manufactured directly from their digital 3D models.

Initial use cases of AM was rapid prototyping for architects and designers (Ngo, Kashani, Imbalzano, Nguyen, & Hui, 2018). But nowadays, AM has several use-cases in the fields such as aeronautical, maritime, turbomachinery, biomedical, spare parts manufacturing, modification of manufactured parts and restoration of broken parts. In the aerospace industry, AM enables engineers to create optimized components with low weight, reduce the manufacturing lead-times and improve but-to-fly ratios (Ding, Shen, Pan, & Cuiuri, 2016). Maritime use cases are including but not limited to afloat manufacturing of spare parts and maintenance of equipment (Strickland, 2016). Complex multi-part components in turbomachinery such as disk-blades and burners can be manufactured as a single part using AM (Klocke et al., 2014). In the field of biomedical applications, AM facilitated creating customized implants, biodegradable implants, etc. (Bartolo et al., 2012).

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A wide range of materials is used in additive manufacturing, and new materials are added to this range continuously. A non-exhaustive list of these materials includes concrete, ceramics, polymers and metals and composites. Concrete is mostly used in building houses using additive manufacturing (Wu, Wang, & Wang, 2016). Among the polymers, acrylonitrile butadiene styrene (ABS) and polylactic acid (PLA) are the most frequently used materials (Gonzalez-Gutierrez et al., 2018). In metal AM, Titanium alloys such as Ti6AL4V, steel alloys such as SS316 or H13, Aluminium alloys such Al-Si-Mg, super alloys such as IN625 and many other alloys are used (Frazier, 2014). Poor mechanical properties of polymers lead to the use of Carbon Fibre Reinforced Plastic (CFRP) in additive manufacturing (Ning, Cong, Qiu, Wei, & Wang, 2015).

In terms of the available standards, American Society for Testing and Materials (ASTM) committee F42 is one of the most active parties in defining standards for Additive manufacturing materials, parts and processes (ASTM, 2018). The European Committee for Standardization (CEN) is also an active organization in the standardization of AM through several actions and projects (CEN-CENELEC, 2018). International Organization for Standardization (ISO) has the ISO/TC 261 committee working on AM, many of them with collaboration with ASME F42 committee (ISO, 2018). The other entity which is active in this field is the National Institute of Standards and Technology (NIST) in the United States (NIST, 2018a). NIST is running multiple projects for supporting standardization of real-time control of additive manufacturing systems, quality assurance AM systems, system integration for AM, and characteristics of additive manufacturing materials (NIST, 2018b). Monzón et al. (2015) reviewed the efforts on developing and implementing standards for AM until 2015.

Although AM brings many advantages to manufacturing, there are some shortcomings as well. Cost of manufacturing with AM relatively high compared to mass production, production is very material and equipment –agnostic and the assuring reliability of the manufactured part is always a big challenge. Although there have been massive invest- ments in the standardization of AM, the process is quite difficult and time-consuming (Jurrens & Energetics Incorporated, 2013; Pellegrino, Makila, McQueen, & Taylor, 2016).

The process of printing a part using AM starts with a digital model 3D of the object. The second step is to add support structures to the part, so that overhanging parts can be printed. Then the model should be cut into slices using slicer software, which replicates the layers which are going to be manufactured (Kathryn et al., 2016).

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2.2.1 Additive manufacturing technologies

In 1986, Charles W. Hull has patented the first method of additive manufacturing called stereolithography or SLA (Charles W. Hull, 1986). Since then, there has been a significant amount of research on the topic. Nowadays, there are several techniques in additive manufacturing namely material extrusion, powder fusion, material jetting, binder jetting, direct energy deposition and sheet lamination etc. In the rest of this subsections, a short description of four of these techniques is provided.

Stereo Lithography (SLA)

In Stereo Lithography (SLA) or Vat Photo-polymerization (VP) a photosensitive liquid monomer, polymer or resin is cured or solidified using a controlled source of ultraviolet light, electron beam or laser. The light applied with the shape of each slice to polymerize the liquid into a solid layer. Then the platform moves downwards to make space for a new layer of liquid of the solidified layer. The process continues until the whole object is shaped layer by layer as shown in Figure 1 (Wong & Hernandez, 2012).

Figure 1. SLM process (Proform, 2018)

Part manufactured by SLM can be post-processed with light curing, to reach to better mechanical properties, and surface enhancement. SLM can be used for manufacturing ceramic parts by adding ceramic particles or using polymer-driven ceramifiable mono- mers (Ngo et al., 2018).

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Possible defects in SLA are shrinkage, curling defect and distortions due to removing the part from the platform. Shrinkage is the direct effect of forming polymers from mon- omers. The curling defect in SLA happens as a result of shrinkage between the layers.

And finally, the process of removing the manufactured part from the platform may cause further distortions in the part due to the liberation of internal forces cumulated between layers (Bugeda, Cervera, Lombera, & Onate, 1995).

Material Extrusion

Material Extrusion AM (MEAM) is the process of softening the material and passing it through a nuzzle and deposit is layer by layer in order to manufacture a 3D part. A MEAM machine usually consists of a two-axis (x and y) CNC manipulator which moves the extruder and a platform which moves in z-axis which moves the manufacturing part downwards to be ready for printing the next layer. The material can be in the form of solid filaments, powders or powder plus bounder liquid and the softening process is normally done by heating. The extrusion process can be done by either plunges, screws or wheels as shown in Figure 2 (Gonzalez-Gutierrez et al., 2018).

Figure 2. Material Extrusion Additive Manufacturing (Gonzalez-Gutierrez et al., 2018)

This technique can be used for manufacturing with metals, polymers, ceramics and composites. In case the material is used as the form of filaments, the process is called Fused Filament Fabrication (FFF) or Fused Deposition Modelling (FDM). FDM is the most common method of AM. FDM machines are available from around one hundred Euros up to several thousand Euros, from desktop home versions up to industrial production versions. The other reason is that the process of manufacturing is safe and simple and the filaments have a good variety of materials (Gonzalez-Gutierrez et al., 2018).

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Cooling profile of FDM manufactured parts have a direct relationship with distortions and porosity in them. The mechanical properties of the parts are affected by the bond between the layers of the manufactured part, which is, in turn, is affected by the temperature of the extruder and the temperature of the last layer of the part (Stavropoulos &

Foteinopoulos, 2018). Poor surface finish and mechanical properties are the main flaws of this technology. Using fibre reinforced filaments can be a solution for the latter problem (Ngo et al., 2018).

Powder Fusion

In powder fusion AM, a thin layer of fine powder which is spread and packed on the top of a descendant platform is fused together using pressure, heat or a binder. The source of the heat can be a laser beam or an electron beam. The fusion process can take place at two levels. In Selective Laser Sintering (SLS) the powder particles are not getting fully melted but they fuse together in molecular level. In a Selective Laser Melting (SLM) or Electron Beam Melting (EBM), the powder particles are melt and the fusion happens in a liquid phase, shown in Figure 3 (Stavropoulos & Foteinopoulos, 2018).

Figure 3. Powder bed fusion (Frazier, 2014)

The quality of parts is highly dependent on the powder shape, size, material and distribution. The other effective parameter is the chemistry and rheology of the binder, in the binder based processes, and the amount and flow of heat energy input to the system in the heat based processes. The heat sintering and melting process cause high residual stress in the manufactured parts. These stresses are the source of several defects in the parts, such as deformations, curling defect, lack of thickness, etc. Therefore, thermal and thermo-mechanical modelling of the process is of utmost importance for optimizing the

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manufacturing process for minimizing the defects (Ngo et al., 2018; Stavropoulos &

Foteinopoulos, 2018). In the next subsection, a detailed description of defects in the SLM process is provided.

Post-processing procedures that usually take place in powder bed techniques are coat- ing, sintering and infiltration. Superior resolution, good surface quality and good mechanical properties of the parts manufactured with powder bed techniques make them one of the most favourable techniques, especially in metal AM (Ngo et al., 2018).

Direct Energy Deposition (DED)

The reason for calling this technique direct energy depositions (DED) that here the energy is guided and focused to a narrow region and the material is deposited and melted simultaneously in the same region. There are several variations for these methods and this technology is mainly used for metal AM. The form of the material feed can be powder or filament and the energy source can be laser, electron beam, or electric arc (Stavropoulos & Foteinopoulos, 2018). Figure 4 is showing a simplified schematic of an electron beam DED.

Figure 4. A generic powder and electron beam DED system (Frazier, 2014) If a DED process uses metal wire filaments and electric arcs, it is called Wire and Arc additive manufacturing (WAAM) (Figure 5). While the powder bed based AM techniques are focused on fine details of the parts, WAAM systems are able to build larger parts (in the scale of 5.8𝑚 × 1.2𝑚 × 1.2𝑚) with higher deposition rates (3 to 10.63 kilograms per hour) (Ding et al., 2016). DED manufacturing systems normally consist of a robotic arm

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and a turning table, therefore they normally have a minimum of five degrees of freedom.

Therefore, it is possible to manufacture parts which are difficult to manufacture with the other technologies. DED is also used for modifying parts and repairing cracks in metal parts (Pinkerton, Wang, & Li, 2008).

Figure 5. Wire and Arc Additive manufacturing (McAndrew et al., 2018)

2.2.2 Defects in Additive Manufacturing

The shape, strength and the size of an AM manufactured part is depending on 1- the raw material used, 2- the manufacturing equipment such as precision of equipment and equipment characteristics, and 3- the process parameters, powder bed temperature, manufacturing environment temperature, such as energy input, nozzle temperature, trav- erse speed, welding torch angle etc. (Kathryn et al., 2016).

Defects in additive manufacturing can be classified into two levels. Defects can cumulate during the manufacturing process and affect the geometry of the part. These defects are normally a result of residual stresses in the workpiece due to the thermal cycle in the manufacturing process, plastic strains caused by shrinkage and constraints of clamping.

Distortions may stop the building process if they become magnificent enough (Mindt et al., 2017).

The other group are defects such as surface roughness, porosity, cracks, splatters and denudation can be described as microscopic defects. For a detailed description of different defects and the factors affecting it in Taheri et al.’s (2017) work.

Defects in Powder Bed Fusion

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In the PBF process with SLM, lack of thickness and curling defect are the two common geometry related defects. Curling defect occurs on the overhang surfaces where parts are not supported with a support structure as shown in Figure 6. The heat conduction rate of an unsupported overhanging part can be up to one hundred times less than a solid material supported part. Employing excessive heat energy, e.g. high laser power, in the layer of an unsupported overhanging part leads to a magnificent thermal constraint on that layer. If this constraint exceeds the strength of the material, a plastic deformation happens. Cumulating these relatively small deformations in multiple layers leads to a curl in the overhanging part (D. Wang, Yang, Yi, & Su, 2013).

Figure 6. Curling defect in overhanging parts (Tounsi & Vignat, 2017)

Curling defect is not purely dependent on the geometry of the part, but also on the choice of the support structure (Tounsi & Vignat, 2017), and process parameter settings (Béraud et al., 2014). Toward reducing this defect, as shown in Figure 7, the support structures are used to dissipate excessive heat and to resist distortion by increasing the inertia of the part. While using more dense support structure seems beneficial to minimize the curling effect, it increases the manufacturing time and material cost (Mokhtarian, Coatanéa, Paris, Mbow, Pourroy, Marin, Vihinen, et al., 2018).

Figure 7. An overhanging part with a support structure (Tounsi & Vignat, 2017)

2.3 Reliability in machinery

Reliability, as De Carlo (2013) defines and discusses, is “the probability that a component (or an entire system) will perform its function for a specific period of time when operating

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in its design environment”. Based on this definition, reliability is a measure for judging that the component is working or not, and the exact environmental and usage conditions should be defined. A broader definition describes reliability as the science to analyse, predict, prevent and mitigates the failures in the time domain.

Failure, or hard failure, is an inoperable state or an event in a system, in which the system or any part of it is not working as specified previously (Dudenhoeffer, 1994). Faults, or soft failures, on the other hand, are the defects which are happening and may or may not cause a failure in the system. Therefore, as shown in Figure 8, failures can be the result of a long term process in which an initial defect escalate among the time and reaches a critical condition that causes the failure in the machine (Lee et al., 2014).

Figure 8. Perception of degradation, diagnostics and prognostics in health manage- ment (Lee et al., 2014)

Failures can be further classified in repairable or non-repairable. In repairable failures, the system can return to its operational state with repair or replacement of a minimal number of system part in a short time. Non-repairable failures are the ones that need the system to be completely replaced or require an extensive overhaul to restore the system (Dudenhoeffer, 1994).

Prognosis and health management

Prognosis and health management (PHM) is an umbrella term which covers many activities in order to maintain the health of a system by diagnosing the faults and taking appropriate decisions based on the prognosis of possible failures. The aim of PHM is to reduce the downtime of the machinery and preventing associated costs.

To create a PHM system, the faults within the system should be identified and the causes for it should be diagnosed. Moreover, the health of a system can be prognosed based on the history of the system and its current situation. The health management discipline assesses the impact of failures and minimizes the possible costs and losses by carrying

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on timely and appropriate maintenance actions based on the output of diagnostics and prognostics (Lee et al., 2014).

To be more precise, diagnosis is detecting the failure mode within a system or among the subsystems. It analyses the nature of a problem and provides the means to isolate it. On the other hand, prognosis tries to indicate the time to the next failure time and remaining useful lifetime of the system until a complete failure occurs. Prognostics continuously uses the indication of degradations in the system and considers the time factor to make the most accurate predictions (Lee et al., 2014).

Fault Diagnosis

To create a fault diagnosis system the essential components are a data collection subsystem to record events and sensor data, a signal processing subsystem to transform sensor data into information and detect faults and a database or knowledge representation system to determine the source of the fault. The knowledge representation subsystem can be implemented using databases, ontologies, physics models, black box models, or Bayesian networks (Lee et al., 2014).

Bayesian networks have been used as a sophisticated tool for creating knowledge representation models for diagnostics in the industrial domain. The possibility of representing uncertainty in the system, expressiveness of BNs, possibility of including expert’s knowledge in the model, modularity and forward and backward simulation are some of the advantages of using BNs in diagnostics. The BN structure can be obtained using expert’s knowledge regarding cause and effects of a failure in the system, mapping already existing models such as fault trees into BNs or using structural learning algorithms which learn the structure from data. A recent literature review on uses of BNs in diagnostics can be found in an article by Cai et al. (2017) work.

Failure Prognosis

On the other hand, prognosis ties to model the degradation of a component and predict the time that a fault or a failure occurs in it. Several methodologies have been developed to create the model and perform the prediction, and described by first hitting time process, remaining useful lifetime (RUL) evaluation, etc. (Letot et al., 2017).

Degradation modes can be classified into normal models, which is estimating the reliability of a model in normal conditions, and accelerated models, which try to estimate the degradation in normal condition given the data obtained in a condition that the time or stress on the component is accelerated (Letot et al., 2017).

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Bayesian networks are very well suited for prognosis uses in reliability engineering. Var- iables which are influencing degradation in equipment, variables related to the operating environment and usage variables are uncertain variables which may have complicated interrelations. The ability to represent dependencies and conditional independencies between variables, efficient calculation scheme, compact representation and interactive interface of Bayesian networks make them a sophisticated tool in fault prognostics (Langseth & Portinale, 2007).

Maintenance

As Letot et al. (2017) describe, maintenance is the act of performing periodic tasks in order to ensure that the functionality of the components is available until the next sched- uled maintenance period.

Several maintenance policies and method have been developed so far, namely corrective maintenance (CM), preventative maintenance (PM), reliability centred maintenance (RCM) and Condition-based maintenance (CBM) etc. Corrective maintenance is the situation in which the equipment is maintained after a failure happens and its purpose is to put the equipment back to the functional state (Peysson, Ouladsine, Noura, Leger, &

Allemand, 2008).

As Lee et al. describe (2014), Preventative maintenance (PM) uses the mean time between failures as a reference for scheduling maintenance for machinery. The strong assumption upon static and deterministic conditions limits this type of maintenance and this method cannot be used under dynamic conditions. PM increases the availability of the system compared to CM and decreases cost up to a tenth the costs of CM (Carlo &

Arleo, 2017), but it is still not optimal for the costs and the time of maintenance. Moreo- ver, the failure history of a system is not the only factor that is effective in predicting the failure time.

On the other hand, for dynamic systems which the future behaviour is not predictable based on the historical observations and the domain knowledge, reliability centred maintenance (RCM) is more suitable. RCM uses statistical tools such as failure modes and effective critically analysis (FMECA) to predict the probability of having expected reliability in a certain period by identifying the failure modes and estimate the time before those failure modes may happen. Nevertheless, RCM is prone to fail if the changes in the dynamics of the system are magnificent.

Condition-based maintenance (CBM) consists of two major activities, data acquisition and condition monitoring. This method is mainly used when the system conditions are

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deterministic, stationary or static and the sensor outputs are a good indicator of the system health.

The other concept in maintenance is the quality of the maintenance. As an alternative to the classical maintenance quality classification described in the literature such as EN 13306:2010, i.e. corrective maintenance and preventative maintenance, a newer classification suggests that a maintenance activity can be perfect, imperfect, minimal, worse or worst, based on the restoration of the equipment after maintenance (Carlo & Arleo, 2017).

De Carlo and Arleo (2017) described these five types of maintenance as follows. A maintenance procedure is called perfect maintenance, when it restores the equipment to an “as good as new” (AGAN) condition. AGAN is a condition in which the maintained equipment would have the same failure rate and lifetime distribution as new equipment and generally is achieved by replacement of all the components with a new one.

Imperfect maintenance renders the equipment to a younger condition, but not to an AGAN condition. The maintained equipment failure rate and lifetime distribution lay somewhere between its premaintenance condition and AGAN condition.

Minimal maintenance restores the equipment just to an “as bad as old” (ABAO) condition, in which, the failure rate and lifetime distribution of the equipment are similar to equipment which has the same age and never failed yet. Minimal repair is done by only replacing faulty components of the equipment. Figure 9 depicts the effect of these three types of maintenance on the failure rate of equipment.

Figure 9. Perfect, imperfect and minimal maintenance and their effect on the failure rate (Carlo & Arleo, 2017)

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Worse maintenance is when the maintenance accidentally causes the equipment to fall into a worse operating condition in terms of failure rate and the lifetime of it. Finally, worst maintenance is the conditions that worse maintenance is accompanied by creating a new failure of breaks in the system.

2.3.1 Characteristics of failure data

The data collected for failures in systems are normally a time series. The data normally consists of readings of several sensors in the system, time stamps for start time, events in the system, maintenance times and failure times (NASA, 2007). Among the sensor data, oil quality and vibration data describe the performance of the machine very well and have been traditionally used for diagnosis purposes. There are several other useful sensor data including, but not limited to, temperature, acoustic emissions, ultrasonic, etc.

The data from several sensors and other sources can be fused together to achieve superior descriptive qualities (Lee et al., 2014).

The process of detection and prediction of failure can be divided into two periods. The first period is the observation interval in which some variables in the system are observed. The second period is the prediction time in which the system is predicting a failure in the future time (Kelleher, Namee, & D’Arcy, 2015). The variables for which the data is recorded in the process of observation can be divided into two groups. Covariates are the variables which represent the characteristics and the environment of the mechanical equipment and response variables are describing the survival times of the equipment (Langseth, 1998).

One of the most important characteristics of the failure data is that this type of data contains censored observations. As Miller et al. (1998) described, data may have four types of censoring. Type one is when the failure in equipment has been observed for a period and the observation is stopped or finished. Then for the equipment which has not failed in that period, there is no failure data recorded, even though it may fail any time after the recording stopped. The second type of censoring is when it is decided to stop recording the failure times after a certain number of failures happened.

The third type of censoring in data happens mostly in medical applications and it’s when the data collection becomes impossible at a random time at the middle of the study. It happens, for example, when the follow up becomes impossible due to patients’ conditions, the patient drops out, etc. It is important to note that for random censoring, a crucial assumption is that the patients are randomly chosen and their type three censored times and their possible failure (decease) time are assumed to be independent.

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Finally, the other type of censoring is interval censoring. For example, if before the be- ginning of the observation, some of the equipment has already experienced failures and there is no record for that it is called left-censored data. If the failures are happening after the data recording stopped, it is called right-censored data and it is similar to type one censoring.

This study tries to provide a brief review of the classical and current methods for fault diagnosis and failure prognosis in section 3.5.1. Afterwards, the methods for creating failure predicting models from single-valued TTF data is reviewed in section 3.5.2. Then a Bayesian network based model for predicting the TTF values and censored TTF values is developed based on a single-valued dataset in section 4.2.

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3. METHODOLOGY

This study takes advantage of several methods from different principles to create mean- ingful models for the systems using experts’ knowledge and data. Bayesian belief networks use several methods from statistics and computer science to obtain the Bayesian network structure from the data, estimate the parameters of the network, perform inference between nodes, etc.

The structure of the Bayesian networks for a problem in a system can also be obtained using the governing equations of the system, domain knowledge, experts’ knowledge and literature. This process can be carried out systematically using dimensional analysis conceptual modelling (DACM) framework which gathers multiple methods from several domains to produce causal graphs between variables of the system and acquire the governing equations between them. The causal graph can be translated into a Bayesian network structure and the governing equations can be used to obtain some of the network’s parameters. To extract experts’ knowledge and use them as parameters for the rest of the nodes, Analytical Hierarchy Process (AHP) from multicriteria decision-making domain is used in this study.

This section also reviews the classical and well-known methods in topics of the case studies. The methods for modelling complex systems in additive manufacturing are reviewed and then a detailed description of the method developed in this study is provided.

In the reliability engineering case study, the classical methods in fault diagnosis and failure prognosis in the field of equipment health management is reviewed. Then the method for creating a predictive model from time to failure datasets with a single value is described.

The rest of this section is formatted as follows. In the first subsection, a detailed description of Bayesian networks and related knowledge and methods that are used in this study is reviewed in section 3.1. Then, in sections 3.2 and 3.3, a brief description of the aspects of the AHP and DACM that are used in this study is provided. And finally, the methods used in the case studies reviewed and the developed methods are described in detail in sections 3.4 and 3.5.

3.1 Bayesian networks

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3.1.1 Backgrounds

To set the ground for a description of the properties and processes in the Bayesian network, it is needed to review the basis of the Bayesian networks theory. In this sub-section, a brief review of Bayesian probabilities, independence between random variables, directed acyclic graphs and causal graphs, the principle of the common cause, Markov causal condition and faithfulness condition, the formal definition of a Bayesian network, d-separation and i-maps is provided.

Probabilistic event and probability distributions

A sample space Ω = {𝜔₁, 𝜔2, … , 𝜔𝑛} for a random procedure is the set of outcomes 𝜔_𝑖, possible for that random procedure. An event 𝐸, which is the phenomenon of interest in probability study, can be defined as a subset of the set Ω. Events in this sense can only have a true/false character. Then, a probability distribution is a function from events space to the space of the real numbers in the range [0,1] and P ∶ ℙ (Ω) → [0,1], in which ℙ (Ω) is called the power set of Ω (Daly et al., 2009).

Since events are subsets of outcomes set, it is possible to use set operations to define the probability of occurrence of two events A and B as 𝑃(𝐴 ∩ 𝐵). Therefore, the conditional probability of occurrence of A, given that event B is occurred is:

𝑃(𝐴|𝐵) = 𝑃(𝐴 ∩ 𝐵)

𝑃(𝐵) (1)

In which 𝑃(𝐵) must be strictly positive. Equation (1) implies that the probability of occurrence of evet 𝐴, given that event 𝐵 is occurred is equal to the joint probability of 𝐴 and 𝐵 divided by the probability of 𝐵. Then intuitively by changing the place of 𝐴 and 𝐵 it can be stated that

𝑃(𝐴|𝐵)𝑃(𝐵) = 𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐵 ∩ 𝐴) = 𝑃(𝐵|𝐴)𝑃(𝐴) (2)

And by rearranging the equation a convenient formula is forming as

𝑃(𝐴|𝐵) =𝑃(𝐴) 𝑃(𝐵|𝐴)

𝑃(𝐵) (3)

which is known as the Bayes’ formula. 𝑃(𝐴) is called prior probability, a priori, or uncon- ditional probability of the event 𝐴. It means the probability of happening of the event 𝐴 without considering any information about event 𝐵. It is also called antecthe edent set of propositions and may lead to consequences when the inference rules are applied to

Bayesian networks in additive manufacturing and reliability engineering

Azarakhsh Hamedi